Question

The trigonometric equation tan^–1x = 3tan^–1 a has solution for ______.

विकल्प

• `|a| ≤ 1/sqrt(3)`

• `|a| > 1/sqrt(3)`

• `|a| < 1/sqrt(3)`

• all real value of a.

Answer 1

The trigonometric equation tan^–1x = 3tan^–1a has solution for `underlinebb(|a| < 1/sqrt(3))`.

Explanation:

To solve the equation tan^–1(x) = 3tan^–1(a), we use the tangent function properties and transformations.

Let θ = tan^–1(a).

Then:

x = tan(3θ)

Using the triple-angle formula for tangent:

tan(3θ) = `(3tan(θ) - tan^3(θ))/(1 - 3tan^3(θ))`

Since tan(θ) = a, substituting a in gives us:

x = `(3a - a^3)/(1 - 3a^2)`

For the function tan⁡⁻¹(x) = 3tan⁻¹(a) to have a solution, the argument of tan (which is 3θ) must be within the range of the tan function, which is `(-π/2, π/2)`.

Therefore, 3θ must also be `(-π/2, π/2)`.

Given that θ = tan⁻¹(a) is within `(-π/2, π/2)`, the condition for 3θ to remain in this interval is:

`-π/6 < θ < π/6`

This translates to:

`-π/6 < tan^-1(a) < π/6`

Taking the tangent of the bounds:

`-1/sqrt(3) < a < 1/sqrt(3)`

Thus, the condition for a is:

`|a| < 1/sqrt(3)`